Curvature correction to vacuum fluctuations and cosmic evolution with cosmological duality

The curvature correction to the Lagrangian of vacuum fluctuations is studied. After a brief tour through the vacuum expectation value of the energy–momentum tensor 〈T _μν 〉 and comparing this with the semiclassical averaging value of T _μν , it is found that vacuum polarization due to curvature determines the effective cosmological constant and the gravitational constant in such a way that the quantum corrections of order R ² are inevitable in the high curvature regime. Considering the asymptotic past and future as vacuum states of the Universe, a Cosmological Duality Conjecture (CDC) is put forward that “if there are terms proportional to R ⁿ (n > 0), there are also terms proportional to R ^?m (m >0) in the gravitational Lagrangian.” The CDC indicates a cyclic cosmological evolution where the late-time dark energy phase comes to an end and a collapsing phase begins with reversal of the arrow of time.     

Regular Rotating Black Holes and Solitons

Solutions describing regular rotating black holes are typically obtained by applying the Newman-Janis complex translation to spherical metrics of the Kerr-Schild class. Regular solutions of this class are specified by T _t ^t = T _r ^r and have necessarily de Sitter centers provided that the source terms T _k ⁱ satisfy the weak energy condition. Rotation transforms a de Sitter center into a de Sitter vacuum disk which is the generic common constituent of all regular rotating objects. In nonlinear electrodynamics coupled to gravity all metrics belong to the Kerr-Schild class and have the Kerr-Newman asymptotic for a distant observer. The ring singularity is replaced with the superconducting ring current which represents an electromagnetic nondissipative source of any asymptotically Kerr-Newman geometry. We outline the basic generic features of regular rotating black holes and solitons which are regular compact objects without event horizons, replacing naked singularities.     

Scalar Field Cosmology in <Emphasis Type="Italic">f</Emphasis>(<Emphasis Type="Italic">R</Emphasis>,<Emphasis Type="Italic">T</Emphasis>) Gravity with Λ

We study the behavior of cosmological parameters, massive and massless scalar fields (normal or phantom) with a scalar potential in f(R, T) theory of gravity for a flat Friedmann-Robertson-Walker (FRW) universe. To get exact solutions to the modified field equations, we use the f(R, T) = R + 2f(T) model by Harko et al. (T. Harko et al., Phys. Rev. D 84, 024020 (2011)), where R is the Ricci scalar and T is the trace of the energy momentum tensor. Our cosmological parameter solutions agree with the recent observational data. Finally, we discuss our results with various graphics.     

Cosmological solutions in low-energy effective field theory for type IIA superstrings

It is known that to obtain cosmological solutions consistent with the Hubble law and accelerated expansion, it is necessary to represent space-time in the form of de Sitter space of the first kind, which ismost simply described by a Robertson-Walkermetric having the scale factor with exponential time dependence. In standard general relativity, this solution is obtained by introduction of the cosmological constant Λ associated with dark energy, whose nature is still not clear. It would be of some interest to consider theories allowing one to obtain cosmological-type solutions without using an effective Λ term. This role was at their time claimed by various supergravity theories which are low-energy approximations of superstring theories. However, it was found that in these theories there are so-called no-go theorems which do not allow solutions in the form of de Sitter space of the first kind. It turned out that this problem can be solved in at least two ways. The first way is to admit the possibility of extra time dimensions, as was demonstrated for the ten-dimensional supergravity by Arefyeva et al. Another way is to consider corrections of higher order in the curvature tensor, which is a subject of this paper. As is known, non-chiral ten-dimensional N = 2 supergravity is a low-energy approximation of type IIA superstring theory. In the present paper, we consider the effective action for type IIA superstring theory, taking into account fourthorder curvature corrections. It is shown that in this case, indeed, one can obtain a cosmological solution in the form of de Sitter space, describing an exponential expansion of space without an effective Λ term.     

Wormholes leading to extra dimensions

In 6D general relativity with a scalar field as a source of gravity, a new type of static wormhole solutions is presented: such wormholes connect our universe with a small 2D extra subspace with a universe where this extra subspace is large, and the whole space-time is effectively 6-dimensional. We consider manifolds with the structure M₀ × M₁ × M₂, where M₀ is 2D Lorentzian space-time while each of M_1,2 can be a 2-sphere or a 2-torus. After selecting possible asymptotic behaviors of the metric functions compatible with the field equations, we give two explicit examples of wormhole solutions with spherical symmetry in our space-time and toroidal extra dimensions. In one example, with a massless scalar field (it is a special case of a well-known more general solution), the extra dimensions have a large constant size at the “far end”; the other example contains a nonzero potential V(φ) which provides a 6D anti-de Sitter asymptotic, where all spatial dimensions are infinite.     

On stable exponential solutions in Einstein–Gauss–Bonnet cosmology with zero variation of <Emphasis Type="Italic">G</Emphasis>

A D-dimensional gravitational model with a Gauss–Bonnet term and the cosmological constant Λ is considered. Assuming diagonal cosmological metrics, we find, for certain Λ > 0, new examples of solutions with an exponential time dependence of two scale factors, governed by two Hubble-like parameters H > 0 and h < 0, corresponding to submanifolds of dimensions m and l, respectively, with (m, l) = (4, 2), (5, 2), (5, 3), (6, 7), (7, 5), (7, 6) and D = 1 + m + l. Any of these solutions describes an exponential expansion of our 3-dimensional factor space with the Hubble parameter H and zero variation of the effective gravitational constant G. We also prove the stability of these solutions in the class of cosmological solutions with diagonal metrics.     

Topologically Nontrivial Solution in Einstein-Dirac Gravity on the Hopf Bundle

A topologically nontrivial solution in Einstein-Dirac gravity with a cosmological constant is obtained. The space-time has the Hopf bundle as a spatial section. It is shown that the Hopf invariant is related to the spinor current density. Two Dirac spinors are used for obtaining a diagonal energy-momentum tensor. Solutions to the nongravitating Dirac equation in background Lorentzian space-time with the Hopf bundle as a spatial section are also obtained. Nongravitating solutions of the Dirac equation are characterized by two quantum half-integer numbers m, n.     

Attracting ellipsoids and synthesis of oscillatory regimes

An approach to stabilization of nonlinear oscillations in multidimensional spaces is proposed on the basis of the V.I. Zubov’s stability theory for invariant sets. As a special case, the derived controls make it possible to excite self-oscillating regimes in specified state subspaces R ^2k ? R ²ⁿ with simultaneous oscillation damping on Cartesian products R ^2n?2k .     

On exponential solutions in the Einstein–Gauss–Bonnet cosmology,stability and variation of <Emphasis Type="Italic">G</Emphasis>

A D-dimensional gravitational model with Gauss–Bonnet and cosmological terms is considered. When an ansatz with a diagonal cosmological metric is adopted, we find new examples of solutions for Λ Λ ≠ 0 and D = 8 with an exponential dependence of the scale factors, which describe expansion of our 3D factor-space and contraction of 4D internal space. We also study the stability of the solutions with static Hubble-like parameters h ⁱ and prove that two solutions with Λ = 0 in dimensions D = 22, 28, which were found earlier, are stable. For both solutions we find asymptotic relations for the effective gravitational constant.     

On risk concentration for convex combinations of linear estimators

We consider the estimation problem for an unknown vector β ∈ Rp in a linear model Y = Xβ + σξ, where ξ ∈ Rⁿ is a standard discrete white Gaussian noise and X is a known n × p matrix with n ≥ p. It is assumed that p is large and X is an ill-conditioned matrix. To estimate β in this situation, we use a family of spectral regularizations of the maximum likelihood method β^α(Y) = H ^α(X ^T X) β ^?(Y), α ∈ R⁺, where β ^?(Y) is the maximum likelihood estimate for β and {H ^α(·): R⁺ → [0, 1], α ∈ R⁺} is a given ordered family of functions indexed by a regularization parameter α. The final estimate for β is constructed as a convex combination (in α) of the estimates β^α(Y) with weights chosen based on the observations Y. We present inequalities for large deviations of the norm of the prediction error of this method.     

Cosmological acceleration from a scalar field and classical and quantum gravitational waves (Inflation and dark energy)

We show that, on the average, a homogeneous and isotropic scalar field and, on the average, homogeneous and isotropic ensembles of classical and quantum gravitational waves generate the de Sitter expansion of empty (with no matter) space-time. At the start and by the end of its cosmological evolution, the Universe is empty. The contemporary Universe is about 70% empty, so the effect of cosmological acceleration should be very noticeable. One can assume that itmanifests itself as dark energy. At the start of the cosmological evolution, before the firstmatter was born, the Universe was also empty. The cosmological acceleration of such empty space-time can manifests itself as inflation. To get the de Sitter accelerated expansion of empty space-time under influence of scalar fields and classical and quantum gravitational waves, one needs to make a mandatory Wick rotation, i.e., one needs to make a transition to Euclidean space of imaginary time. One can assume that the very existence of inflation and dark energy could be considered as a possible observable evidence for the fact that time by its nature could be a complex value which manifests itself precisely at the start and by the end of the evolution of the Universe, i.e., in those periods when the Universe is empty (or nearly empty).     

On horizons and wormholes in k-essence theories

We study the properties of possible static, spherically symmetric configurations in k-essence theories with the Lagrangian functions of the form F(X), X ≡ ?,_α ?,^α. A no-go theorem has been proved, claiming that a possible black-hole-like Killing horizon of finite radius cannot exist if the function F(X) is required to have a finite derivative dF/dX. Two exact solutions are obtained for special cases of kessence: one for F(X) = F ₀ X ^1/3, another for F(X) = F ₀|X|^1/2 ? 2Λ, where F ₀ and Λ are constants. Both solutions contain horizons, are not asymptotically flat, and provide illustrations for the obtained nogo theorem. The first solution may be interpreted as describing a black hole in an asymptotically singular space-time, while in the second solution two horizons of infinite area are connected by a wormhole.     

Optimisation of a non linear fuzzy function

 We present a first study concerning the optimization of a non linear fuzzy function f depending both on a crisp variable and a fuzzy number: therefore the function value is a fuzzy number. More specifically, given a real fuzzy number ?∈F and the function f(a,x):R ²→R, we consider the fuzzy extension induced by f, f˜ : F × R → F, f˜(?,x) = Y˜. If K is a convex subset of R, the problem we consider is “maximizing”f˜(?,x), xˉ∈ K. The first problem is the meaning of the word “maximizing”: in fact it is well-known that ranking fuzzy numbers is a complex matter. Following a general method, we introduce a real function (evaluation function) on real fuzzy numbers, in order to get a crisp rating, induced by the order of the real line. In such a way, the optimization problem on fuzzy numbers can be written in terms of an optimization problem for the real-valued function obtained by composition of f with a suitable evaluation function. This approach allows us to state a necessary and sufficient condition in order that ∈K is the maximum for f˜ in K, when f(a,x) is convex-concave (Theorem 4.1).     

Constraints on scalar coupling to electromagnetism

We review a possible nonminimal (dilatonic) coupling of a scalar field (axion-like particle) to electromagnetism, through experimental and observational constraints. Such a coupling is motivated from recent quasar spectrum observations that indicate a possible spatial and/or temporal variation of the fine-structure constant. We consider a dilatonic coupling of the form B _F (?) = 1+g?. The strongest bound on the coupling parameter g is derived from the weak equivalence principle tests, which impose g < 1.6 × 10^?17 GeV^?1. This constraint is strong enough to rule out this class of models as a cause for an observable cosmological variation of the fine structure constant unless a chameleon mechanism is implemented. Also, we argue that a similar coupling occurs in chameleon cosmology, another candidate dark mater particle, and we estimate the cosmological consequences by both effects. It should be clarified that this class of models is not necessarily ruled out in the presence of a chameleon mechanism which can freeze the dynamics of the scalar field in high density laboratory regions.     

Redshift in the model of embedded spaces

The non-configurational geometrization of the electromagnetic field can be realized using the Model of Embedded Spaces (MES). This model assumes the existence of proper 4D space-time manifolds of particles with a nonzero rest mass and declares that physical space-time is the metric result of the dynamic embedding of these manifolds: the value of the partial contribution of the element manifold is determined by element interactions. The space of the model is provided with a Riemann-like geometry, whose differential formalism is described by a generalization of the gradient operator ?/?x ⁱ → ?/?x ⁱ + 2u ^k ?²/?x[ ⁱ ?u ^k ], where u ⁱ = dx ⁱ /ds is a matter velocity. In the paper, the redshift effect existing in the space of MES is considered, and its electromagnetic component is analyzed. It is shown that for cold matter of the modern Universe this component reduces to a shift in electric fields and is described by the expression \(\Delta {\omega _e}/\omega \simeq \mp \sqrt k \Delta {\varphi _e}/{c^2} = \mp 0.861 \cdot {10^{ - 21}}\Delta {\varphi _e}\left( V \right)\), where the potential is measured in volts and the sign must be determined experimentally. Testing of the effect is the “experimentum crusis” for MES.     

ΛCDM type Heckmann–Schuking model and Union 2.1 compilation

We study a ΛCDM type cosmological model in Heckmann-Schucking space-time, by using 287 high redshift (.3 ≤ z ≤ 1.4) SN Ia data on observed absolute magnitude along with their possible error from Union 2.1 compilation. We use the χ² test to compare Union 2.1 compilation observed data and the corresponding theoretical values of the apparent magnitude (m). It is found that the best fit value of (Ω _m )₀, (Ω_Λ)₀ and (Ω _σ )₀ are 0.2940, 0.7058 and 0.0002, respectively, and the derived model represents the features of an accelerating universe which is consistent with the recent astrophysical observations.     

Inflationary scenario in Bianchi Type V space-time for a barotropic fluid distribution with variable bulk viscosity and vacuum energy density

We examine an inflationary scenario in Bianchi Type V space-time for a barotropic fluid distribution with variable bulk viscosity and decaying vacuum energy density. We observe that the matter density ρ, the coefficient of bulk viscosity ζ and the expansion θ all diverge at τ = 0. The spatial volume increases with time, representing an inflationary scenario. The deceleration parameter q < 0 for barotropic, dust and radiation dominated models representing an accelerated universe, while for a stiff fluid distribution q > 0 corresponding to a decelerated universe. The vacuum energy density Λ decreases with time. The entropy per unit volume is proportional to the absolute temperature. The energy conditions (weak, dominant and strong) are discussed for the model. The reality condition ρ + p ≥ 0 is violated for the inflationary model due to the presence of a scalar field (φ). We also discuss the importance of Bianchi Type V model where the anisotropy dies away during the inflationary era. We also calculate the inflationary parameters and compare the results with the Planck data and discuss their compatibility with anisotropy and BAO estimates. The cosmological constant Λ is a function of time without break general covariance. We also discuss the bounds of the model, how the model isotropizes, where the fluid goes after inflation and how viscosity may realize a graceful exit from inflation to a radiation dominated era.     

Dyonic configurations in nonlinear electrodynamics coupled to general relativity

We consider static, spherically symmetric configurations in general relativity, supported by nonlinear electromagnetic fields with gauge-invariant Lagrangians depending on the single invariant f = F_μνF^μν. After a brief review on black hole (BH) and solitonic solutions, obtained so far with pure electric or magnetic fields, an attempt is made to obtain dyonic solutions, those with both electric and magnetic charges. A general scheme is suggested, leading to solutions in quadratures for an arbitrary Lagrangian function L(f) (up to some monotonicity restrictions); such solutions are expressed in terms of f as a new radial coordinate instead of the usual coordinate r. For the truncated Born-Infeld theory (depending on the invariant f only), a general dyonic solution is obtained in terms of r. A feature of interest in this solution is the existence of a special case with a self-dual electromagnetic field, f ≡ 0 and the Reissner-Nordström metric.     

Solvability conditions of problem of synthesizing proportional-integral control of a linear MIMO system

The solvability conditions and just the solution of the problem of the regular and irregular proportional-integral (PI) control are found in accordance with the properties of invariant zeros of a multi-input multioutput (MIMO) system. It is proved that the problem of synthesizing the control of the MIMO system is solvable if and only if the pair of matrices (A, B) that describes a control plant is controllable and the matrix B_L^⊥AC_R^⊥ (where B_L^⊥ is the left zero divisor of the matrix B and C_R^⊥ is the right zero divisor of the output matrix C) has a complete row rank.     

Property-Preserving Data Reconstruction

We initiate a new line of investigation into online property-preserving data reconstruction. Consider a dataset which is assumed to satisfy various (known) structural properties; e.g., it may consist of sorted numbers, or points on a manifold, or vectors in a polyhedral cone, or codewords from an error-correcting code. Because of noise and errors, however, an (unknown) fraction of the data is deemed unsound, i.e., in violation with the expected structural properties. Can one still query into the dataset in an online fashion and be provided data that is always sound? In other words, can one design a filter which, when given a query to any item I in the dataset, returns a sound item J that, although not necessarily in the dataset, differs from I as infrequently as possible. No preprocessing should be allowed and queries should be answered online.We consider the case of a monotone function. Specifically, the dataset encodes a function f:{1,…,n}?? R that is at (unknown) distance ε from monotone, meaning that f can—and must—be modified at ε n places to become monotone.Our main result is a randomized filter that can answer any query in O(log?² nlog? log?n) time while modifying the function f at only O(ε n) places. The amortized time over n function evaluations is O(log?n). The filter works as stated with probability arbitrarily close to 1. We provide an alternative filter with O(log?n) worst case query time and O(ε nlog?n) function modifications. For reconstructing d-dimensional monotone functions of the form f:{1,…,n} ^d ? ? R, we present a filter that takes (2 ^O(d)(log?n)^4d?2log?log?n) time per query and modifies at most O(ε n ^d ) function values (for constant d).